This is a fast applet to compare the effectiveness of various methods of Riemann sums.[br]You can change the function and the number of subintervals.

It should be noted that the left and right hand rules are exact for constant functions.[br]The midpoint and trapezoid rules are exact for linear functions.[br]Simpson's rule is exact for cubic functions.[br][br]In general you should not how much the error is diminished by doubling the number of intervals.